Dynamical Systems · Interactive

Phase Portraits & Equilibria

What kind of equilibrium a system has, read straight off the eigenvalues of its Jacobian.

This is a toolkit I built from the labs I taught as an LA for LS 30B (Math for Life Scientists). Near an equilibrium, any 2-D system acts like a linear one, x' = Ax, and the eigenvalues of A tell you the type: node, spiral, saddle, or center. Edit the matrix below, click the plane to release trajectories, and watch the classification and the trace–determinant marker update live.

Linearization & the Jacobian

\mathbf{x}' = \mathbf{f}(\mathbf{x}) \;\approx\; J(\mathbf{x}^{*})\,(\mathbf{x}-\mathbf{x}^{*}), \qquad J = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial y} \\[8pt] \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial y} \end{bmatrix}

The eigenvalues of the Jacobian are the roots of its characteristic polynomial, so the trace and determinant alone fix them:

\lambda^{2} - (\operatorname{tr}J)\,\lambda + \det J = 0

Classification (Lab 7)

eigenvalues	type
complex, negative real part	stable spiral
complex, positive real part	unstable spiral
complex, zero real part	center
real, both negative	stable node
real, both positive	unstable node
real, opposite signs	saddle

Beyond equilibria: chaos

The last LS 30B labs went past fixed points and cycles into chaos. Romeo, Juliet, and the Nurse form a love triangle whose equations are secretly the Lorenz system. Slide r up and the story changes: a settled fixed point, then a limit cycle, then full chaos near r = 28, where two near-identical starts diverge completely. The logistic map gets there a different way — period-doubling as its growth rate rises.